Determine whether the equation $y = 5x^2 + 2x$ is proportional or not.

1. Problem Description

Determine whether the equation $y = 5x^2 + 2x$ is proportional or not.

2. Solution Steps

A proportional relationship can be expressed in the form $y = kx$, where $k$ is the constant of proportionality. In a proportional relationship, $y$ is directly proportional to $x$.

The given equation is $y = 5x^2 + 2x$. We can see that the equation has a term with $x^2$ and a term with $x$. For the equation to represent a proportional relationship, it should be in the form $y=kx$. The equation given is not in that form. If we tried to force it into the $y=kx$ form, we would have

$k = \frac{y}{x} = \frac{5x^2 + 2x}{x} = 5x + 2$.

Since $k$ depends on $x$, it is not a constant.

Another way to think about this is that a proportional relationship must pass through the origin $(0,0)$. Let's plug in $x=0$ into the equation:

$y = 5(0)^2 + 2(0) = 0$

The equation passes through the origin. However, this is not enough to determine if it is proportional.

Let's consider two different values of $x$.

If $x=1$, $y = 5(1)^2 + 2(1) = 5+2 = 7$.

If $x=2$, $y = 5(2)^2 + 2(2) = 5(4) + 4 = 20+4 = 24$.

For the relationship to be proportional, $\frac{y}{x}$ must be constant.

In the first case, $\frac{y}{x} = \frac{7}{1} = 7$.

In the second case, $\frac{y}{x} = \frac{24}{2} = 12$.

Since the ratio is not constant, the equation is not proportional.

3. Final Answer

Non-Proportional